I am working on a written (essay) assignment regarding the properties of continuous functions on a closed interval [a,b]. In my case, I chose to discuss this set of theorems, but I am having trouble understanding one in specific. Graphically I can follow it along. I believe I have a pretty good grasp on Theorem 1 (f(a) < 0 < f(b) means we must have an f(x) = 0), Theorems 4 and 5 (if f(a) < c < f(b), then we must have f(x) = c. Theorem 5 is the same but with f(a) > c > f(b)).
The theorem I am having trouble understanding has to do with what are called Theorem 10/11. Specifcally, I am looking at Theorem 11, which is an extension of theorem 10. Here is Theorem 10:
This is more the part I am having trouble understanding. From the graphical analysis, it seems like it's manipulating an interval (or choosing an interval) such that when we look for f(y) <= f(x), , that f(y) <= f(x) for the entire line, not just at a point (because there could be a function where y0 isn't necessarily the lowest point on the entire line, just on an interval) . Is that a correct way of describing what theorem 10 entails?
Now, here is theorem 11:
To me, Theorem 11 is saying something that I can graphically show easily, but am having trouble describing just in plain English. Seems just to show that up to a certain point, if we take the function of a number below a constant c, that we will be guaranteed to have a solution to the equation and if c < m, it fails.
Is there a better way to describe Theorem 10 and 11? As always, thanks again in advance, you guys are the best!